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Decomposition method for solving fractional Riccati differential equations. (English) Zbl 1107.65121

Summary: We implement a relatively new analytical technique, the Adomian decomposition method, for solving fractional Riccati differential equations. The fractional derivatives are described in the Caputo sense. In this scheme, the solution takes the form of a convergent series with easily computable components. The diagonal Padé approximants are effectively used in the analysis to capture the essential behavior of the solution. The corresponding solutions of the integer order equations are found to follow as special cases of those of fractional order equations. Some numerical examples are presented to illustrate the efficiency and reliability of the method.

MSC:

65R20 Numerical methods for integral equations
45J05 Integro-ordinary differential equations
26A33 Fractional derivatives and integrals
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[1] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press New York · Zbl 0918.34010
[2] Podlubny, I., Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calculus Appl. Anal., 5, 367-386 (2002) · Zbl 1042.26003
[4] He, J. H., Some applications of nonlinear fractional differential equations and their approximations, Bull. Sci. Technol., 15, 2, 86-90 (1999)
[5] He, J. H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Methods Appl. Mech. Engrg., 167, 57-68 (1998) · Zbl 0942.76077
[6] Grigorenko, I.; Grigorenko, E., Chaotic dynamics of the fractional Lorenz system, Phys. Rev. Lett., 91, 3, 034101-034104 (2003)
[7] Mainardi, F., Fractional calculus: ‘Some basic problems in continuum and statistical mechanics’, (Carpinteri, A.; Mainardi, F., Fractals and Fractional Calculus in Continuum Mechanics (1997), Springer-Verlag: Springer-Verlag New York), 291-348 · Zbl 0917.73004
[8] Anderson, B. D.; Moore, J. B., Optimal Control-Linear Quadratic Methods (1990), Prentice-Hall: Prentice-Hall New Jersey · Zbl 0751.49013
[9] Diethelm, K.; Ford, J. M.; Ford, N. J.; Weilbeer, W., Pitfalls in fast numerical solvers for fractional differential equations, J. Comput. Appl. Math., 186, 482-503 (2006) · Zbl 1078.65550
[12] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), John Wiley and Sons Inc.: John Wiley and Sons Inc. New York · Zbl 0789.26002
[13] Oldham, K. B.; Spanier, J., The Fractional Calculus (1974), Academic Press: Academic Press New York · Zbl 0428.26004
[14] Caputo, M., Linear models of dissipation whose \(Q\) is almost frequency independent. Part II, J. Roy. Astral. Soc., 13, 529-539 (1967)
[15] Adomian, G., A review of the decomposition method in applied mathematics, J. Math. Anal. Appl., 135, 501-544 (1988) · Zbl 0671.34053
[16] Adomian, G., Solving Frontier Problems of Physics: The Decomposition Method (1994), Kluwer Academic Publishers: Kluwer Academic Publishers Boston · Zbl 0802.65122
[17] Shawagfeh, N.; Kaya, D., Comparing numerical methods for the solutions of systems of ordinary differential equations, Appl. Math. Lett., 17, 323-328 (2004) · Zbl 1061.65062
[18] Shawagfeh, N. T., Analytical approximate solutions for nonlinear fractional differential equations, Appl. Math. Comput., 131, 2, 517-529 (2002) · Zbl 1029.34003
[19] Momani, S., An explicit and numerical solutions of the fractional KdV equation, Math. Comput. Simul., 70, 2, 110-118 (2005) · Zbl 1119.65394
[20] Momani, S., Non-perturbative analytical solutions of the space- and time-fractional Burgers equations, Chaos Solitons & Fractals, 28, 4, 930-937 (2006) · Zbl 1099.35118
[21] Momani, S.; Odibat, Z., Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method, Appl. Math. Comput., 177, 2, 488-494 (2006) · Zbl 1096.65131
[22] Boyd, J., Padè approximants algorithm for solving nonlinear ordinary differential equation boundary value problems on an unbounded domain, Comput. Phys., 11, 3, 299-303 (1997)
[23] Momani, S., Analytical approximate solutions of nonlinear oscillators by the modified decomposition method, Int. J. Mod. Phys. C, 15, 967-979 (2004) · Zbl 1119.65356
[24] Wazwaz, A. M., Analytical approximations and Padé approximants for Volterra’s population model, Appl. Math. Comput., 100, 13-25 (1999) · Zbl 0953.92026
[25] Cherruault, Y., Convergence of Adomian’s method, Kybernetes, 18, 31-38 (1989) · Zbl 0697.65051
[26] Cherruault, Y.; Adomian, G., Decomposition methods: a new proof of convergence, Math. Comput. Model., 18, 103-106 (1993) · Zbl 0805.65057
[27] Baker, G. A., Essentials of Padè Approximants (1975), Academic Press: Academic Press London · Zbl 0315.41014
[28] Burden, R. L.; Faires, J. D., Numerical Analysis (1993), PWS: PWS Boston · Zbl 0788.65001
[29] El-Tawil, M. A.; Bahnasawi, A. A.; Abdel-Naby, A., Solving Riccati differential equation using Adomian’s decomposition method, Appl. Math. Comput., 157, 2, 503-514 (2004) · Zbl 1054.65071
[30] Abbasbandy, S., Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian’s decomposition method, Appl. Math. Comput., 172, 485-490 (2006) · Zbl 1088.65063
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